Solving Systems Of Equations When X Is Known Finding The Ordered Pair

Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving systems of equations. We'll walk through it step by step, ensuring you not only understand the solution but also grasp the underlying concepts. So, let's get started!

Unraveling the Mystery: Solving for the Ordered Pair

In this mathematical quest, we're presented with a classic scenario: solving a system of equations. Shawn, our math whiz, has already done some groundwork, finding that x = 3. Our mission, should we choose to accept it, is to determine the ordered pair that satisfies the following system:

2x + 4y = 34
6x + 2y = 32

An ordered pair, as you might recall, is simply a pair of numbers, typically written as (x, y), that represents a point on a coordinate plane. In the context of systems of equations, the ordered pair that solves the system is the point where the lines represented by the equations intersect. This point's coordinates, the x and y values, make both equations true simultaneously. Now, knowing that x = 3 is half the battle, let's leverage this information to find the corresponding y value and, ultimately, the ordered pair solution.

The core concept here is substitution. Since we already know the value of x, we can substitute it into either of the given equations to solve for y. Let's start with the first equation, 2x + 4y = 34. Replacing x with 3, we get 2(3) + 4y = 34, which simplifies to 6 + 4y = 34. To isolate y, we subtract 6 from both sides, resulting in 4y = 28. Finally, we divide both sides by 4 to find that y = 7. So, based on the first equation, the ordered pair solution appears to be (3, 7). But before we pop the champagne, let's verify this solution using the second equation.

Now, let's use the second equation, 6x + 2y = 32, to confirm our solution. Again, we substitute x with 3, giving us 6(3) + 2y = 32, which simplifies to 18 + 2y = 32. Subtracting 18 from both sides, we get 2y = 14. Dividing both sides by 2, we find that y = 7. Lo and behold, we arrive at the same y value! This confirms that our ordered pair (3, 7) is indeed the solution to the system of equations. This methodical approach of substituting the known value of x into both equations and solving for y not only gives us the solution but also provides a crucial check to ensure our answer is correct. It's like having a mathematical safety net!

Therefore, the ordered pair that solves the system of equations is (3, 7). We've successfully navigated this mathematical maze by employing the power of substitution and verification. Remember, guys, when faced with systems of equations, breaking down the problem into smaller, manageable steps, like substituting known values and checking solutions, can make even the most daunting challenges feel conquerable.

Delving Deeper: Why Substitution Works

Now that we've found our solution, let's take a moment to appreciate the why behind the method. Substitution isn't just a magic trick; it's a logical process rooted in the fundamental principles of algebra. At its core, a system of equations represents a set of relationships between variables. The goal is to find the values of these variables that satisfy all the relationships simultaneously. When we're given that x = 3, we're essentially handed a vital piece of the puzzle. We know that in the solution, the x-coordinate must be 3.

Therefore, wherever we see x in our equations, we can confidently replace it with 3, as this replacement preserves the truth of the equation in the context of the solution. This is the heart of substitution: we're replacing a variable with its known value to create a simpler equation with only one unknown, y in our case. Think of it like swapping a placeholder with the real deal. Once x is replaced with 3, the equations transform into statements about y alone. This allows us to isolate y using standard algebraic techniques, such as adding, subtracting, multiplying, and dividing both sides of the equation. It's like narrowing down a suspect in a detective story – each step brings us closer to the final answer.

The beauty of substitution lies in its versatility. It's a powerful tool that can be applied to a wide range of systems of equations, regardless of their complexity. Whether you're dealing with linear equations, quadratic equations, or even more exotic types, the principle remains the same: replace a known variable with its value to simplify the system. However, to fully appreciate the elegance of substitution, it's essential to grasp the underlying algebraic principles. Equations, after all, are statements of equality. Any operation we perform on one side of an equation must be mirrored on the other side to maintain this balance. This is why we can add or subtract the same value from both sides, or multiply or divide both sides by the same non-zero value, without altering the solution. These operations are the bedrock of algebraic manipulation, allowing us to isolate variables and ultimately solve for their values. So, guys, the next time you're tackling a system of equations, remember that substitution is your trusty sidekick, ready to help you unravel the mysteries of the mathematical world.

Alternative Paths: Exploring Other Solution Methods

While substitution was our method of choice for this problem, it's worth noting that systems of equations can be solved using various approaches. One popular alternative is the elimination method, which, as the name suggests, involves eliminating one of the variables by manipulating the equations. In our case, we could multiply the first equation by -3, resulting in -6x - 12y = -102. Then, we could add this modified equation to the second equation (6x + 2y = 32). Notice that the x terms cancel out, leaving us with -10y = -70. Dividing both sides by -10, we find that y = 7, which aligns with our previous result. To find x, we would then substitute y = 7 into either of the original equations.

The elimination method is particularly useful when the coefficients of one of the variables are opposites or multiples of each other. This allows for easy cancellation through addition or subtraction. However, if the coefficients are not so conveniently related, we might need to multiply one or both equations by carefully chosen constants to set up the elimination. This might seem like more work than substitution in some cases, but the choice of method often comes down to personal preference and the specific structure of the equations.

Another way to visualize and solve systems of equations is graphically. Each equation represents a line on the coordinate plane. The solution to the system is the point where these lines intersect. By graphing the equations 2x + 4y = 34 and 6x + 2y = 32, we would find that they intersect at the point (3, 7), confirming our algebraic solution. Graphing can be a powerful tool for understanding the concept of solutions to systems of equations, but it might not be the most precise method for finding exact solutions, especially if the intersection point has non-integer coordinates. Ultimately, the most effective approach is to have a diverse toolbox of solution methods – substitution, elimination, and graphing – and to choose the one that best suits the problem at hand. Remember, guys, math is about flexibility and problem-solving agility!

Real-World Connections: Systems of Equations in Action

Systems of equations might seem like an abstract mathematical concept, but they have a surprising number of real-world applications. From economics to engineering, these systems are used to model and solve problems involving multiple variables and constraints. For example, imagine a business trying to determine the optimal pricing strategy for two products. They might have equations representing the demand for each product as a function of its price, as well as equations representing the costs of production and distribution. By solving this system of equations, the business can find the prices that maximize profit.

In physics, systems of equations are used to analyze the motion of objects under multiple forces. For instance, consider a projectile launched at an angle. Its trajectory is influenced by gravity and air resistance. By setting up equations representing these forces and solving them simultaneously, physicists can predict the projectile's range, maximum height, and time of flight. This is crucial for designing everything from artillery to sporting equipment.

In computer science, systems of equations play a role in areas like computer graphics and game development. When rendering 3D scenes, computers need to perform complex calculations to determine how light interacts with objects and how they appear from different viewpoints. These calculations often involve solving systems of equations to model the geometry and lighting of the scene.

Even in everyday life, we encounter situations that can be modeled using systems of equations. Suppose you're planning a party and need to buy drinks and snacks. You have a budget and know the prices of each item. You can set up a system of equations to determine the optimal quantities of each item to buy, ensuring you stay within your budget and satisfy your guests. So, guys, the next time you're faced with a problem involving multiple unknowns and relationships, remember that systems of equations might just be the key to unlocking the solution.

Wrapping Up: Mastering Systems of Equations

And there you have it! We've successfully navigated the world of systems of equations, solving for the ordered pair solution using the substitution method. We've also explored alternative solution methods like elimination and graphing, and even touched upon the real-world applications of these powerful mathematical tools. Hopefully, this journey has not only equipped you with the skills to solve similar problems but has also sparked an appreciation for the elegance and versatility of systems of equations. Remember, guys, math isn't just about memorizing formulas; it's about understanding the underlying concepts and developing a problem-solving mindset. So, keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is vast and fascinating, and there's always something new to discover!